由半立方体II产生的同调表示
Homology representations arising from the half cube, II
摘要：
在以前的工作（arXiv：0806.1503v2），我们定义了一个家庭的$n$ -维半立方体的子复形，通过删除内部的所有半立方体形状的面的尺寸至少$k$，我们证明了这样的子复形的同源性是集中在度$k-1$。该同源基团支持$D$型Coxeter基团$W(D_n)$的天然作用。在本文中，我们明确确定的字符（在${\Bbb C}$），这些同调表示，原来是多重自由。作为对称群$S_n$的表示，通过限制，同调表示是由抛物子群导出的某些表示的直和。$\sym_n$的后一种表示（在${\Bbb C}$上）与$\sym_n$在$k$ -等实超平面排列的补的$(k-2)$ -nd同调上的表示一致。
英文摘要：
In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes of the $n$-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least $k$, and we proved that the homology of such a subcomplex is concentrated in degree $k-1$. This homology group supports a natural action of the Coxeter group $W(D_n)$ of type $D$. In this paper, we explicitly determine the characters (over ${\Bbb C}$) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group $S_n$ by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of $\sym_n$ agree (over ${\Bbb C}$) with the representations of $\sym_n$ on the $(k-2)$-nd homology of the complement of the $k$-equal real hyperplane arrangement.
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